  ABnBAn  BAn  30)(and,18)–(,7)–( ,then
 BAn  

The symbols and equations you provided seem to be related to set theory and involve the union and intersection of sets \( A \) and \( B \). Let's decipher and solve this step-by-step.

Given:
1. \( |A \cup B| = 30 \) (the union of sets \( A \) and \( B \) has 30 elements),
2. \( |A| = 18 \) (set \( A \) has 18 elements),
3. \( |B| = 7 \) (set \( B \) has 7 elements).

We are tasked to find \( |A \cap B| \), the number of elements in the intersection of sets \( A \) and \( B \).

### Formula for Union of Two Sets:
The general formula for the union of two sets is:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

### Substituting the known values:
\[
30 = 18 + 7 - |A \cap B|
\]
\[
30 = 25 - |A \cap B|
\]
\[
|A \cap B| = 25 - 30 = -5
\]

However, the intersection of two sets cannot have negative elements. This suggests that there may be an inconsistency or misunderstanding in the problem formulation. Could you confirm or clarify the values provided, particularly whether the numbers given for \( |A| \), \( |B| \), or \( |A \cup B| \) are correct?

Do you need further clarification on this or any other part of the problem? Here are some related questions:

1. Can you verify the set sizes for \( A \) and \( B \)?
2. Would you like to explore different types of set operations?
3. How does the formula for set unions and intersections work in general cases?
4. Would you like to discuss examples of sets with negative elements?
5. Do you want help in drawing Venn diagrams for better understanding?

**Tip:** Always check that the sum of individual set sizes and the total union size make sense in real-world scenarios.

The query submitted by the user: ABnBAn = BAn = 30)(and,18)–(,7)–( ,then  BAn  =

Explore how to solve a set theory problem involving the union and intersection of two sets with given sizes. Understand the application of the union formula and its implications in set theory. Suitable for students in Grades 9-12.

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